a) Ta có: \(a+b+c=0\)
\(\Rightarrow2abc\left(a+b+c\right)=0\)
\(\Rightarrow2a^2bc+2ab^2c+2abc^2=0\)
Ta lại có:
\(a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)^2\) (cái này bạn tự chứng minh nha)
\(\Rightarrow a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2+4a^2bc+4ab^2c+4abc^2\)
\(\Rightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\right)\)
\(\Rightarrow a^4+b^4+c^4=2\left(ab+bc+ca\right)^2\left(đpcm\right)\)
b) Ta có: \(a+b+c=0\)
\(\Rightarrow a=-\left(b+c\right)\)
\(\Rightarrow a^2=b^2+c^2+2bc\)
\(\Rightarrow a^2-b^2-c^2=2bc\)
\(\Rightarrow a^4+b^4+c^4-2a^2b^2-2a^2c^2+2b^2c^2=4b^2c^2\)
\(\Rightarrow a^4+b^4+c^4=4b^2c^2+2a^2b^2+2a^2c^2-2b^2c^2\)
\(\Rightarrow a^4+b^4+c^4=2a^2b^2+2a^2c^2+2b^2c^2\)
\(\Rightarrow a^4+b^4+c^4+a^4+b^4+c^4=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2\)
\(\Rightarrow2\left(a^4+b^4+c^4\right)=\left(a^2+b^2+c^2\right)^2\)
\(\Rightarrow a^4+b^4+c^4=\frac{\left(a^2+b^2+c^2\right)^2}{2}\left(đpcm\right)\)
Chúc bạn học tốt và tíck cho mìk vs nhé!
